Introducing money. Olivier Blanchard April 2002 14.452. Spring 2002. Topic 6.
14.452. Spring, 2002 2 No role for money in the models we have looked at. Implicitly, centralized markets, with an auctioneer: ² Possibly open once, with full set of contingent markets. (Remember, no heterogeneity, no idiosyncratic shocks. (Arrow Debreu) ² More appealing. Markets open every period. Spotmarkets,basedonexpectationsofthefuture. Forexample,market for goods, labor, and one{period bonds. A sequence of temporary equilibria (Hicks). Still no need for money. An auctioneer. Some clearing house. So need to move to an economy where money plays a useful role. The ingredients. ² No auctioneer. Geographically decentralized trades. ² Then, problem of double coincidence of wants. Barter is not convenient. Money, accepted on one side of each transaction, is much more so. Two types of questions Foundations ² Why money? What kind of money will emerge? ² Can there be competing monies? ² Fiat versus commodity money? ² Numeraire versus medium of exchange? Should they be the same, or not?
14.452. Spring, 2002 3 Not just abstract, or history. The rise of barter in Russia in the 1990s. \Units of ac- \Natural" dollarization in some Latin American countries. count" in Latin America. But most of the time, we can take it as given that money will be used intransactions,thatitwillbe atmoney,andthatthenumeraireandthe medium of exchange will be the same. If we take these as given, then we can ask another set of questions: ² How di erent does a decentralized economy with money look like? ² What determines the demand for money, the equilibrium price level, nominal interest rates? ² How does the presence of money a ect the consumption/saving choice? ² Steady state and dynamic e ects of changes in the rate of money growth. Start by looking at a benchmark model. Cash in advance. Then, look at variations on the model; money in the utility function. Then focus on price and in ation dynamics, especially hyperin ation. 1 A cash in advance model Think in terms of a decentralized economy (although we shall see that there is an optimization problem which replicates the outcome). 1.1 The optimization problem of consumers/workers Consumers/workers maximize: i=1 X E[ i=0 iu(c t+i j t ]
14.452. Spring, 2002 4 subject to: and C t + M t+1 + B t+1 = W t + t + M t +(1+i t )B t + X t M t C t Note that I ignore: ² Uncertainty Because it is not central to the points I want to make. But there is no problem in introducing it in the usual way. ² The labor/leisure choice. Itwouldbea ected. ButIleaveitout for simplicity. People supply one unit of labor inelastically. The notation: is the price of goods in terms of the numeraire (the price level). M t and B t are holdings of money and bonds at the start of period t. W t and t are the nominal wage and nominal pro t received by each consumer respectively. i t is the nominal interest rate (the interest rate stated in dollars, not goods) paid by the bonds. X t is a nominal transfer from the government (which has to be there if and when we think of changes in money as being implemented by distribution of new money to consumers). Now turn to the assumptions underlying the speci cation: ² Consumers care only about consumption. They do not derive utility from money. ² The rst constraint is the budget constraint. It says that nominal consumption plus new asset holdings must be equal to nominal income wage income (the labor supply is inelastic and equal to one) and pro t
14.452. Spring, 2002 5 income plus initial asset holdings, including interest on the bonds, plus nominal government transfers. ² If the only constraint was the rst constraint, then people would hold no money: Bonds pay interest, money does not. The second constraint explains why people hold money. It is known as the cash in advance (CIA) constraint. People must enter the period with enough nominal money balances to pay for consumption. ² One story here. People are composed of a worker and a consumer. The worker goes to work. The consumer goes to buy goods, and must do this before the worker has been paid. So he must have su±cient money balances to nance consumption. ² One can think of more sophisticated, smoother, formulations. For example: The cost of buying consumption goods is decreasing in money balances. I shall return to this below. Let t+i i be associated with the budget constraint, ¹ t+i i be associated with the CIA constraint. Set up the Lagrangian and derive the FOC. C t : U 0 (C t )=( t + ¹ t ) M t+1 : t = ( t+1 + ¹ t+1 ) B t+1 : t = (1 + i t+1 ) t+1 Interpretation of each. Can combine them to get: U 0 (C t ) = [ (1 + i t+1 )] U 0 (C t+1 ) 1+i t +1 1+i t+1
14.452. Spring, 2002 6 Note that =+1 =1+¼ t+1. If we de ne the real interest rate as: (1 + r t+1 ) Pt (1 + i t+1 ) +1 We can rewrite the rst order condition as: Interpretation. U 0 (C t ) 1+i t = (1 + r t+1 ) U 0 (C t+1 ) 1+i t+1 ² Because people have to hold money one period in advance, the e ective price of consumption is not 1 but 1 + i. ² Once we adjust for this price e ect, then we get the same old relation, between marginal utility this period, marginal utility next period, and the real interest rate. Note the role of both the nominal and the real interest rates. Note that the nominal interest rate is constant, the equation reduces to the standard Euler equation: U 0 (C t )= (1 + r t+1 )U 0 (C t+1 ) This characterizes consumption. Consumption behavior is very similar to that in the non monetary economy. Two di erences: ² The relative price e ect, if i t is di erent from i t+1. ² The fact that the rate of return on total wealth is lower (as some of wealth does not yield interest), so the feasible level of consumption is lower. Given consumption, the characterization of the demand for money is straightforward. The CIA holds as an equality:
14.452. Spring, 2002 7 M t = C t Pure quantity theory. No interest rate elasticity. Simple, but possibly too simple. Will look at extensions below. 1.2 The optimization problem of rms Firms produce goods using labor and capital. They pay labor a wage W t. They buy capital for use in the next period, and they nance these purchases of capital by issuing nominal bonds. Their nominal cash ow is thus given by: t = F (K t ;N t ) W t N t (1 + i t )B t + (1 ±)K t K t+1 + B t+1 Cash ow is equal to production minus the wage bill, minus payment of interest and principal on bonds issued last period, plus the value of the remaining capital stock minus the value of the capital purchased, plus bond issues. The value of a rm is given by the present value of nominal cash ow, discounted by the relevant nominal interest rate. V t = t +(1+i t+1 ) 1 V t+1 The three FOC for rms are given by: N t : F N (K t ;N t )=W t B t+1 : 1 = 1
14.452. Spring, 2002 8 K t+1 : =(1+i t+1 ) 1 [+1 (1 ± + F K (K t+1 ;N t+1 ))] Note the second FOC: It says that the amount of bonds issued by rms is irrelevant. They could nance purchases of capital from current pro t, or partly through bond issues, or fully through bond issues. Their decisions would be the same. (But, under our assumption, there are nominal bonds in the economy, which makes it easier to think about the nominal interest rate). The third FOC can be rewritten as: Or: (1 ± + F K (K t+1 ;N t+1 )) = (1 + i t+1 ) +1 (1 ± + F K (K t+1 ;N t+1 )) = (1 + r t+1 ) Firms purchase capital to the point where the marginal product of capital is equal to the real interest rate. 1.3 The equilibrium and the steady state To close the model, we have that: N t =1 Turn to the government budget constraint. Assume that the stock of money is changed through transfers to people: X t = M t+1 M t Putting things together, the dynamics of the economy are characterized by the following equations:
14.452. Spring, 2002 9 U 0 (C t ) 1+i t = (1 + r t+1 ) U 0 (C t+1 ) 1+i t+1 (1 + i t )=(1+r t )(1 + ¼ t ) (1 + r t )=1 ± + F K (K t ; 1) M t = C t K t+1 = F (K t ; 1) + (1 ±)K t C t I shall not attempt to look at dynamics, but just focus on steady state: Suppose that the rate of growth of nominal money is equal to x, so. X t =(1+x 1) M t = x M t In steady state, C t ;K t ;r t ;i t ;¼ t are constant, so: From the FOC of the consumer, and the demand for capital by rms: (1 + r) =1+F K (K; 1) ± =1= This is the same rule as without money: The modi ed golden rule. In steady state, real money balances must be constant, so: ¼ = x In ation is equal to money growth. And so, i = ¼ + r = x + r. Thisone for one e ect of money growth on the nominal interest rate is known as the Fisher e ect.
14.452. Spring, 2002 10 Using these relations in the budget constraint of the consumer gives: C = F (K; 1) ±K So, on the real side, the economy looks the same as before. In addition people hold money. And in ation proceeds at the same rate as money growth. The fact that, in steady state, money growth has no e ect on the real allocation is refered to as the superneutrality of money. Is this superneutrality a general result? I now explore an alternative formalization. 2 Money in the utility function The CIA constraint is too tight. One can clearly maintain a lower level of real money balances is one is willing to go to the ATM machine more often. More reasonable to assume that ² The higher the level of real money balances one holds, the lower the transaction costs, so the higher the level of output net of transaction costs, ² Or the higher the level of utility, again net of transaction costs. One can formalize this explicitly, A dynamic Baumol Tobin model. This is what is done by Romer (see original article or BF). Very useful, but a bit heavy for here. One can take short cuts. Real money balances in the production function, or in the utility function. See e ects of putting money in the utility function. (Sidrauski model). So the optimization problem of consumers/workers is: E[ X iu(c t+i ; M t+i +i ) j t ]
14.452. Spring, 2002 11 subject to: C t + M t+1 + B t+1 = W t + t + M t +(1+i t )B t + X t where, plausibly U m > 0andU mc 0(why?). Let t+i i be the lagrange multiplier associated with the constraint. Then the FOC are given by: C t : U c (C t ; M t )= t B t+1 : t = t+1 (1 + i t+1 ) M t+1 : t = t+1 + 1 U m (C t+1 ; M t+1 ) +1 +1 Interpretation. Can rewrite as: An intertemporal condition: U c (C t ; M t )= (1 + r t+1 )U c (C t+1 ; M t+1 ) +1 An intratemporal condition U m (C t ; M t )=U c (C t ; M t )= i t Interpretation. Note that the second says that the ratio of marginal utilities has to be equal to the opportunity cost of holding money, so i, the nominal interest rate. If for example, U(C; M=P) = log(c)+a log(m=p)
14.452. Spring, 2002 12 Then, M t =(a= ) C t i t This gives us an LM relation. (Indeed you can think of the rst condition as giving us a simple IS relation, this giving us an LM relation. More on this in the next lectures). The demand for money is a function of the level of transactions, here measured by consumption, and the opportunity cost of holding money, i. Turn to steady state implications. ( rms' side is the same as before). 1+r =1= C = F (K; 1) ±K U m (C; M P )=U c(c; M P )= (x + r) So, same real allocation again. And a level of real money balances inversely proportional to the rate of in ation, itself equal to the rate of money growth. Dynamic e ects? Yes. But nothing very exciting. Can make it more exciting by modelling trips to the bank and having people come at di erent times. Then, distribution e ects. But does not seem to capture much of what we actually observe. So, bottom line: Money as a medium of exchange, without nominal rigidities gives us a way of thinking about the economy, the price level, the nominal interest rate, but not much in the way of explaining uctuations. Very useful however when money growth and in ation become high and variable. Turn to this.
14.452. Spring, 2002 13 3 Money growth, in ation, seignorage Start with the money demand we just derived: M t = C t L(r t + ¼t e ) If money growth and in ation are high and variable, M, P and ¼ e will move a lot relative to C and r. So assume, for simplicity, that C t = C, and r t = r, so: M t = CL(r + ¼t e ) This gives a relation between the price level and the expected rate of in ation. The higher expected in ation, the lower real money balances, the higher the price level. This relation, together with an assumption about money growth, and the formation of expectations, allows us to think about the behavior of in ation. This is what Cagan did. Looking at hyperin ations, he asked; ² Was hyperin ation the result of money growth, and only money growth? ² Why was money growth so high? Did it maximize seignorage. And if not, then why? Now have a quick look at his model (Read the paper, written in 1956. It is a great read, even today). Also, read BF4-7, and BF10-2. What follows is just a sketch. Continuous time, more convenient here. Assume a particular form for the demand for money: So, in logs: M=P =exp( ¼ e )
14.452. Spring, 2002 14 m p = ¼ e Log real money balances are a decreasing function of expected in ation. Or di erentiating with respect to time: x ¼ = d¼ e =dt Assume that people have adaptive expectations about expected in ation. (In an environment such as hyperin ation, this assumption makes a lot of sense. More on rational expectations below). d¼ e =dt = (¼ ¼ e ) Money growth and in ation Suppose money growth is constant, at x. Will in ation converge to ¼ = x? To answer, combine the two equations above and eliminate d¼ e =dt between the two, to get: x ¼ = (¼ ¼ e ) This is a line in the (¼; ¼ e )space. Foragivenx, d¼ e =d¼ = (1 )=, so if < 1 the line is downward sloping. If > 1 upward sloping. ² If < 1, then the equilibrium is stable. Start with x>0, and ¼ =0. Then converge to ¼ = ¼ e = x. ² If > 1, then not. Why? Cagan estimated and, found < 1. Hyperin ation was the result of money growth, not a bubble. Seignorage
14.452. Spring, 2002 15 What is the maximum revenue the government can get from money creation (called seignorage: S dm=dt P So, in steady state: = dm=dt M M P = x exp( ¼e ) S = x exp( x) So x =1= Much lower than the growth rates of money observed during hyperin ation. But just a steady state result. Can clearly get more in the short run, when ¼ e has not adjusted yet. This suggests looking at di erent dynamics: Given seignorage, dynamics of money growth and in ation. Seignorage, money growth and in ation Start from: S = x exp( ¼ e ) For a given S, draw the relation between ¼ e and x in ¼ e ;x space. Concave. Can cross the 45 degree line twice, once if tangent, not at all if no way to generate the required seignorage in steady state. Which equilibrium is stable? Using the equation for adaptive expectations and the money demand relation in derivative form: Or: d¼ e =dt = (¼ ¼ e )= (x + d¼ e =dt ¼ e ) d¼ e =dt =1=(1 ) (x ¼ e )
14.452. Spring, 2002 16 If two equilibria, lower one is stable. Start from it, and suppose S increases so no equilibrium. Then, money growth and in ation will keep increasing. This appears to capture what happens during hyperin ations. Some other issues ² Adaptive or rational expectations? (see BF 5-1) ² Fiscal policy, and the e ects of in ation on the need for seignorage. (See Dornbusch et al) ² Unpleasant monetarist arithmetic? (see BF 10-2) From Cagan: Seven Hyperin ations of the 1920s and 1940s Country Beginning End P T =P 0 Average Monthly Average Monthly In ation rate (%) Money Growth (%) Austria Oct. 1921 Aug. 1922 70 47 31 Germany Aug. 1922 Nov. 1923 1.0x10 10 322 314 Greece Nov. 1943 Nov. 1944 4.7x10 6 365 220 Hungary 1 Mar. 1923 Feb. 1924 44 46 33 Hungary 2 Aug. 1945 Jul. 1946 3.8x10 27 19,800 12,200 Poland Jan. 1923 Jan. 1924 699 82 72 Russia Dec. 1921 Jan. 1924 1.2x10 5 57 49 P T =P 0 : Price level in the last month of hyperin ation divided by the price level in the rst month.